A class of differential equations of PI-type with the quasi-Painlevé property. (English) Zbl 1150.34031
The paper is concerns with the equation of the form
\[
y'' = \frac{2(2k+1)}{(2k-1)^2}y^{2k}+x
\]
which is the first Painlevé equation for \(k=1\). The author shows that the equation admits the quasi-Painlevé property and admits no entire solution. Moreover, every solution is transcendental. If \(k \geq 2\), the equation admits no meromorphic solution and admits a two-parameter family of solutions which are at least \(\nu\)-valued for any positive integer \(\nu\).
Reviewer: Yuefei Wang (Beijing)
MSC:
| 34M55 | Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies |
| 34M05 | Entire and meromorphic solutions to ordinary differential equations in the complex domain |
Keywords:
Nonlinear differential equation; Quasi-Painlev’e property; Painlev’e equation; Hyperelliptic integralReferences:
| [1] | Boutroux, P., Recherches sur les transcendantes de M. Painlevé et l’étude asymptotique des équations différentielles du second ordre, Ann. Sci. École Norm. Sup., 30, 255-375 (1913) · JFM 44.0382.02 |
| [2] | Gromak, V. I.; Laine, I.; Shimomura, S., Painlevé Differential Equations in the Complex Plane (2002), Berlin: Walter de Gruyter, Berlin · Zbl 1043.34100 |
| [3] | Hayman, W., Meromorphic Functions (1964), Oxford: Clarendon Press, Oxford · Zbl 0115.06203 |
| [4] | Hille, E.: Analytic Function Theory, vol. 2. Chelsea, New York (1973) · Zbl 0273.30002 |
| [5] | Hille, E., Ordinary Differential Equations in the Complex Domain (1976), New York: Wiley, New York · Zbl 0343.34007 |
| [6] | Hinkkanen, A.; Laine, I., Solutions of the first and second Painlevé equations are meromorphic, J. Anal. Math., 79, 345-377 (1999) · Zbl 0986.34074 |
| [7] | Hurwitz, A.; Courant, R., Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen (1925), Berlin: Springer, Berlin · JFM 51.0236.12 |
| [8] | Ince, E. L., Ordinary Differential Equations (1956), New York: Dover, New York · Zbl 0063.02971 |
| [9] | Kimura, H., The degeneration of the two dimensional Garnier system and the polynomial Hamiltonian structure, Ann. Mat. Pura Appl., 155, 25-74 (1989) · Zbl 0693.34043 · doi:10.1007/BF01765933 |
| [10] | Kimura, T., Sur les points singuliers essentiels mobiles des équations différentielles du second ordre, Comment. Math. Univ. Sancti Pauli, 5, 81-94 (1956) · Zbl 0071.30902 |
| [11] | Kirwan, F., Complex Algebraic Curves (1992), Cambridge: Cambridge University Press, Cambridge · Zbl 0744.14018 |
| [12] | Kitaev, A. V., Elliptic asymptotics of the first and the second Painlevé transcendents, Russ. Math. Surv., 49, 81-150 (1994) · Zbl 0829.34040 · doi:10.1070/RM1994v049n01ABEH002133 |
| [13] | Laine, I.: Nevanlinna Theory and Complex Differential Equations. Studies in Mathematics, vol. 15. Walter de Gruyter, Berlin, New York (1993) · Zbl 0784.30002 |
| [14] | Okamoto, K., Studies on the Painlevé equations, I. Ann. Mat. Pura Appl., 146, 337-381 (1987) · Zbl 0637.34019 · doi:10.1007/BF01762370 |
| [15] | Painlevé, P.: Œ uvres de Paul Painlevé, tome 1. Centre Nat. Recherches Sci., Paris (1972) |
| [16] | Shimomura, S., Pole loci of solutions of a degenerate Garnier system, Nonlinearity, 14, 193-203 (2001) · Zbl 1011.34077 · doi:10.1088/0951-7715/14/2/301 |
| [17] | Shimomura, S., Proofs of the Painlevé property for all Painlevé equations, Jpn. J. Math., 29, 159-180 (2003) · Zbl 1085.34071 |
| [18] | Steinmetz, N., On Painlevé’s equations I, II and IV, J. Anal. Math., 82, 363-377 (2000) · Zbl 0989.34073 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
